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2021 | OriginalPaper | Chapter

# Computing Rational Points on Rank 0 Genus 3 Hyperelliptic Curves

Authors : María Inés de Frutos-Fernández, Sachi Hashimoto

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## Abstract

We compute rational points on genus 3 odd degree hyperelliptic curves C over $$\mathbb {Q}$$ that have Jacobians of Mordell–Weil rank 0. The computation applies the Chabauty–Coleman method to find the zero set of a certain system of p-adic integrals, which is known to be finite and include the set of rational points $$C(\mathbb {Q})$$. We implemented an algorithm in Sage to carry out the Chabauty–Coleman method on a database of 5870 curves.
Footnotes
1
The global height is the absolute logarithmic height of the point, which is the maximum of the absolute logarithmic heights of its coordinates. For a rational point $$\frac {n}{d}$$, this height is computed as $$\max (\log (|n|),\log (|d|)).$$

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Title
Computing Rational Points on Rank 0 Genus 3 Hyperelliptic Curves
Authors
María Inés de Frutos-Fernández
Sachi Hashimoto
2021
DOI
https://doi.org/10.1007/978-3-030-80914-0_14